Phase averaging of the coherent head-on beam–beam instability

Within the framework of a simplifying model and within the first approximation of the perturbation theory we discuss the effects of the phase averaging on the stability of the coherent oscillations of identical, flat and head-on colliding bunches. Near the resonances n/(2m), where n and m are integers, these oscillations are unstable. Finite lengths of the colliding bunches can result in the phase averaging suppression of increments of unstable modes. Both betatron and synchro-betatron oscillation modes can be excited in the collisions of such bunches. The coupling of these modes due to the beam–beam interactions changes the stability conditions of coherent oscillations of the colliding bunches.